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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

An epistemic extension of equilibrium logic and its relation to Gelfond’s epistemic specifications

Andreas Herzig

University of Toulouse and CNRS, IRIT, France (joint work with Luis Fari˜ nas del Cerro and Ezgi Iraz Su; paper @ IJCAI 2015)

Dagstuhl, May 2015

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Outline

1

Answer-set programming and Gelfond’s epistemic specifications

2

HT models and equilibrium models

3

Epistemic HT models and equilibrium models

4

Comparison by examples

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Answer-set programming (ASP)

logic program = set of rules rules of form Head ← Body Body may contain “not”

‘default negation’, ‘negation by failure’ = non-deducibility

semantics: not every classical model of a program intended Π = {p ← not q} should have unique model {p} Π = {p ← not p} should have no model Π = {p ← p} should have unique model ∅

(all variables false)

consensus only in the 90ies: answer set semantics

fixed point definition:

V is an answer set for Π iff V = reduct(Π, V)

nonmonotonic inference relation remarkably stable: 10+ different characterisations

[Lifschitz ”Twelve Definfitions of a Stable Model”, ICLP 08]

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Answer-set programming (ASP)

logic program = set of rules rules of form Head ← Body Body may contain “not”

‘default negation’, ‘negation by failure’ = non-deducibility

semantics: not every classical model of a program intended Π = {p ← not q} should have unique model {p} Π = {p ← not p} should have no model Π = {p ← p} should have unique model ∅

(all variables false)

consensus only in the 90ies: answer set semantics

fixed point definition:

V is an answer set for Π iff V = reduct(Π, V)

nonmonotonic inference relation remarkably stable: 10+ different characterisations

[Lifschitz ”Twelve Definfitions of a Stable Model”, ICLP 08]

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

ASP lacks expressivity

Example (scholarship eligibility program [Gelfond 94])

1

eligible ← highGPA

2

eligible ← fairGPA, minority

3

eligible ← fairGPA, highGPA

4

interview ← not eligible, not eligible

5

highGPA or fairGPA ←

has the answer sets AS(Πeligible) =

• {highGPA, eligible},

{fairGPA}

• Therefore:

Πeligible | interview ⇒ wanted: quantification over (possible) answer sets. . .

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

ASP lacks expressivity

Example (scholarship eligibility program [Gelfond 94])

1

eligible ← highGPA

2

eligible ← fairGPA, minority

3

eligible ← fairGPA, highGPA

4

interview ← not eligible, not eligible

5

highGPA or fairGPA ←

has the answer sets AS(Πeligible) =

• {highGPA, eligible},

{fairGPA}

• Therefore:

Πeligible | interview ⇒ wanted: quantification over (possible) answer sets. . .

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Epistemic specifications [Gelfond 91, 94]

Example (scholarship eligibility program, ES-version)

1

eligible ← highGPA

2

eligible ← minority, fairGPA

3

eligible ← fairGPA, highGPA

4

interview ← not K eligible, not K eligible

5

highGPA or fairGPA ←

will have the answer sets AS(ΠK eligible) =

• {highGPA, eligible, interview},

{fairGPA, interview}

• Therefore:

ΠK eligible | ≈ interview

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Epistemic specifications [Gelfond 91, 94]

Example (scholarship eligibility program, ES-version)

1

eligible ← highGPA

2

eligible ← minority, fairGPA

3

eligible ← fairGPA, highGPA

4

interview ← not K eligible, not K eligible

5

highGPA or fairGPA ←

will have the answer sets AS(ΠK eligible) =

• {highGPA, eligible, interview},

{fairGPA, interview}

• Therefore:

ΠK eligible | ≈ interview

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Epistemic specifications: language

rules of form Head ← Body

Body may contain “not” Body may contain epistemic operators K q = “it is known that q” M q = “q may be believed”

models:

1

move from answer sets to world views = sets of answer sets

2

reduct ΠW of an epistemic specification Π by a world view W ⇒ eliminates the epistemic operators ⇒ procedural

3

W is a world view of Π iff W = AS(ΠW)

⇒ fixpoint equation ⇒ non-constructive

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Epistemic specifications: reduct definition [Kahl 14]

Definition (reduct ΠW of an epistemic specification Π by a world view W)

for each rule, literal in body: if true in W: if false in W:

K l

replace by l delete rule

not K l

replace by ⊤ replace by not l

M l

replace by ⊤ replace by not not l

not M l

replace by not l delete rule examples:

{p ← K p}{∅} = ∅ {p ← K p}{{p}} = {p ← p} {p ← M p}{∅} = {p ← not not p} {p ← M p}{{p}} = {p ← ⊤}

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Outline

1

Answer-set programming and Gelfond’s epistemic specifications

2

HT models and equilibrium models

3

Epistemic HT models and equilibrium models

4

Comparison by examples

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Which logical account of ASP?

logic behind negation by failure?

identify “not” with “¬”

problem: classical negation doesn’t suit

program {p ← not p} should have no model but equivalent to p in classical logic

⇒ logic of here-and-there (HT)

minimisation of truth in HT models

⇒ equilibrium models

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

The logic of here-and-there (HT)

H•

• T
• simple Kripke models:
• nly two possible worlds H (‘here’) and T (‘there’)

accessibility relation is reflexive, and T is accessible from H idea: H = proved true, T = hypothesised, PVAR \ T = refuted intuitionistic heredity condition: H ⊆ T ⊆ PVAR

language: connective ‘→’

stronger than material ‘⊃’: |= ¬ϕ ↔ (ϕ→⊥) |= ϕ → ¬¬ϕ |= ϕ ← ¬¬ϕ |= ϕ ∨ ¬ϕ |= (¬ϕ → ϕ) ↔ ϕ

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

The logic of here-and-there (HT)

HT model = couple (H, T) such that H ⊆ T ⊆ PVAR

H = T: ‘total model’

truth conditions: H, T |= p iff p ∈ H H, T |= ¬ϕ iff T, T |= ϕ H, T |= ϕ→ψ iff H, T |= ϕ ⊃ ψ and T, T |= ϕ ⊃ ψ

(where ⊃ is material implication)

Theorem (strong equivalence [Lifschitz et al. 01])

AS(Π1 ∪ Π) = AS(Π2 ∪ Π) for every Π, iff |=HT Π1 ↔ Π2

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Equilibrium models = minimal HT models

Definition (T, T) equilibrium model of ϕ iff

1

T, T |= ϕ

2

H, T |= ϕ for every H ⊂ T

Theorem ([Pearce 96])

equilibrium models of Π = answer sets of Π

⇒ t.b.d.: epistemic extension to capture epistemic specifications

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Outline

1

Answer-set programming and Gelfond’s epistemic specifications

2

HT models and equilibrium models

3

Epistemic HT models and equilibrium models

4

Comparison by examples

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Epistemic equilibrium logic

in a nutshell:

1

combine HT and S5

semantics: sets of HT models intermediate modal logic (stronger than intuitionistic S5)

2

maximise falsehood as in equilibrium logic: ∅ | ≈EE K ¬p

p∨q | ≈EE K (p ∨ q)

however:

p∨q | EE M p ∧ M q

3

maximise ignorance:

p∨q | ≈AEE M p ∧ M q

⇒ Autoepistemic [Moore 80] ⇒ “all-that-I-know” [Levesque 90]

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Epistemic HT-models

Definition

EHT model = (W, ) where

W is a classical S5 model: W ⊆ 2PVAR : W −→ 2PVAR such that (T) ⊆ T for every T ∈ W

classical S5 model: = id truth conditions:

(W, ), T |= p iff p ∈ (T) (W, ), T |= ϕ → ψ iff (W, ), T |= ϕ ⊃ ψ and (W, id), T |= ϕ ⊃ ψ (W, ), T |= K ϕ iff (W, ), T ′ |= ϕ for every T ′ ∈ W (W, ), T |= M ϕ iff (W, ), T ′ |= ϕ for some T ′ ∈ W

satisfies the requirements for intuitionistic modal logics

[Fischer-Servi 76, Fari˜ nas&Raggio 83, Simpson 95, . . . ]

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Epistemic equilibrium models

minimise truth (cf. equilibrium logic)

Definition W is an epistemic equilibrium model of ϕ iff

1

W, id |= ϕ

(classical S5 model of ϕ)

2

there is no id such that W, |= ϕ (no ‘weaker’ EHT-model of ϕ) Example: K p → p has unique epistemic eq.model

• ∅
• Example: p ∨ q has 3 epistemic eq.models:
• {p}
• ,
• {q}
• , and
• {p}, {q}
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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Autoepistemic equilibrium models

minimise knowledge (cf. Levesque’s “all-that-I-know”)

Definition W is an autoepistemic equilibrium model of ϕ iff

1

W is an epistemic equilibrium model of ϕ

2

no W′ s.th. W ⊂ W′ is an epistemic equilibrium model of ϕ

(no ‘bigger’ epi.eq.model of ϕ)

3

no W′ s.th. W <ϕ W′ is an epistemic equilibrium model of ϕ

(no ‘better’ epi.eq.model of ϕ)

where W ≤ϕ W′ iff . . .

Example: {p or q ←} has single autoepistemic eq.model

• {p}, {q}
• Theorem (strong equivalence)

...

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Outline

1

Answer-set programming and Gelfond’s epistemic specifications

2

HT models and equilibrium models

3

Epistemic HT models and equilibrium models

4

Comparison by examples

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Example: cycle with K

Π18 =

• p ← K p
• has two world views
• ∅
• and
• {p}
• [Gelfond 91, 94; Wang&Zhang 05]

has unique world view

• ∅
• [Gelfond 11; Kahl 14; FHS 15]
• Remark. clear case: K p → p is the truth axiom of epistemic logic

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Example: cycle with M

Π1 =

• p ← M p
• has 2 world views
• ∅
• and
• {p}
• [Gelfond 91; 94; Wang&Zhang 05]

has unique world view

• {p}
• [Kahl 14; FHS 15]
• Remark. circular, but corresponds to intuitions

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Example: preference over a disjunction

Π32 =

• p or q ←,

q ← M p

• has no world view

[Gelfond 91; 94; 11]

has unique world view

• {q}
• [Kahl 14; FHS 15]
• Remark. intuitively clear (similar to Gelfond’s eligibility example)

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Example: another preference over a disjunction

Π32 =

• p or q ←,

q ← not K p

• has 2 world views
• {q}
• and
• {p}
• [Gelfond 91; 94; 11]

has unique world view

• {q}
• [Kahl 14; FHS 15]
• Remark. intuitively clear (similar to Gelfond’s eligibility example)

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

Example: a difference with Kahl

Π32 =

• p ← not K not q, not q ,

q ← not K not q, not q

• ?? has 2 world views
• ∅
• and
• {p, q}
• [Kahl 14]

? has unique world view

• {p, q}
• [FHS 15]
• Remark. our solution seems to better match intuitions

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ASP and epistemic specifications HT models and equilibrium models Epistemic HT models and equilibrium models Comparison

To sum it up

zoo of semantics for epistemic specifications

best candidate: [Kahl 14] arguably flawed: [Gelfond 91, 94; Wang&Zhang 05] problem with preference over disjunction: [Gelfond 05]

in the examples, our epistemic HT almost matches [Kahl 14] good basis for further work:

simple intuitionistic modal logic epistemic equilibrium models (minimise truth) autoepistemic equilibrium models (minimise knowledge)

related work: [Wang and Zhang 05]

but matches [Gelfond 94]

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